Seminar @ KEK
Jun. 9, 2009
On gravity duals for NRCFTs
Dept. of Phys. Kyoto Univ.
Kentaroh Yoshida
Based on the works,
Gravity
Sean Hartnoll, K.Y, arXiv:0810.0298,
Sakura Schäfer Nameki, Masahito Yamazaki, K.Y,
arXiv:0903.4245
1
1.
INTRODUCTION
AdS/CFT correspondence
Quantum gravity, non-perturbative definition of string theory
Gravity (string) on AdS space
CFT
Application of classical gravity to strongly coupled theory
Application of AdS/CFT:
quark-gluon plasma (hydrodynamics)
condensed matter systems (superfluidity)
EX.
superconductor
quantum Hall effect
[Gubser, Hartnoll-Herzog-Horowitz]
[Davis-Kraus-Shar] [Fujita-Li-Ryu-Takayanagi]
Most of condensed matter systems are non-relativistic (NR)
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NR limit in AdS/CFT
Gravity (string) on AdS space
CFT
? ? ?
NRCFT
What is the gravity dual ?
NR conformal symmetry
Schrödinger symmetry
EX fermions at unitarity
Today
Let us discuss gravity solutions preserving Schrödinger symmetry
(or other NR scaling symmetry)
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Plan of the talk
1.
Introduction (finished)
2.
Schrödinger symmetry
3.
Coset construction of Schrödinger spacetime
[S.Schafer-Nameki-M.Yamazakil-K.Y]
4.
String theory embedding
[Hartnoll-K.Y]
5.
Summary and Discussion
4
2. Schrödinger symmetry
5
What is Schrödinger algebra ?
Non-relativistic analog of the relativistic conformal algebra
Conformal
Poincare
Schrödinger algebra
Galilei
[Hagen, Niederer,1972]
= Galilean algebra + dilatation + special conformal
Dilatation
(in NR theories)
dynamical exponent
EX
Free Schrödinger eq.
(scale inv.)
6
Special conformal trans.
a generalization of mobius tras.
The generators of Schrödinger algebra
= Galilean algebra
C has no index
7
The Schrödinger algebra
Galilean algebra
(Bargmann alg.)
Dilatation
Dynamical exponent
Special conformal
SL(2) subalgebra
8
Algebra with arbitrary z
Galilean algebra
+
Dilatation
Dynamical exponent
•
M is not a center any more.
•
conformal trans. C is not contained.
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FACT
A Schrödinger algebra in d+1 D is embedded into
a ``relativistic’’ conformal algebra in (d+1)+1 D as a subalgebra.
EX.
Schrödinger algebra in 2+1 D can be embedded into SO(4,2) in 3+1 D
This is true for arbitrary z.
A relativistic conformal algebra in (d+1)+1 D
The generators:
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The embedding of the Schrödinger algebra in d+1 dim. spacetime
(Not contained for z>2)
LC combination:
A light-like compactification of Klein-Gordon eq.
(For z=2 case)
(d+1)+1 D
KG eq.
with
Sch. eq.
d+1 D
The difference of dimensionality
Rem:
This is not the standard NR limit of the field theory
Remember the light-cone quantization
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Application of the embedding to AdS/CFT
CFT
The field theory is compactified on the light-like circle:
DLCQ description
NRCFT = LC Hamiltonian
Gravity
with
-compactification
[Goldberger,Barbon-Fuertes ]
Symmetry is broken from SO(2,d+2) to Sch(d) symmetry
But the problem is not so easy as it looks.
What is the dimensionally reduced theory in the DLCQ limit?
The DLCQ interpretation is applicable only for z=2 case.
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3.
Coset construction of
Schrödinger spacetime
[Sakura Schafer-Nameki, M. Yamazaki, K.Y., 0903.4245]
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There may be various Schrödinger inv. gravity sols.
other than the DLCQ of AdS backgrounds.
Deformations of the AdS space with
-compactification
preserving the Schrödinger symmetry
(degrees of freedom of deformation)
Schrödinger spacetime
AdS space
[Son, Balasubramanian-McGreevy]
deformation term
This metric satisfies the e.o.m of Einstein gravity with a massive vector field
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Coset construction of Schrödinger spacetime
[S. Schafer-Nameki,
M. Yamazaki, K.Y., 0903.4245]
A homogeneous space can be represented by a coset
EX
: isometry,
: local Lorentz symmetry
We want to consider degrees of freedom to deform the AdS metric
within the class of homogeneous spacetime by using the coset construction.
As a matter of course, there are many asymptotically Schrödinger inv. sol.
but we will not discuss them here.
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Coset construction of the metric
vielbeins
1.
spin connections
MC 1-from
vielbeins
2.
Contaction of the vielbeins:
symm. 2-form
If G is semi-simple
straightforward
(Use Killing form)
But if G is non-semi-simple, step 2 is not so obvious. (No non-deg. Killing form)
NOTE:
MC 1-form is obviously inv. under left-G symm. by construction.
The remaining is to consider right-H inv. at step 2.
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Nappi-Witten’s argument
[Nappi-Witten, hep-th/9310112]
2D Poincare with a central extension
Killing form
(degenerate)
P1
P2
J
T
right-G inv.
Most general symmetric 2-form
The condition for the symm. 2-form
PP-wave type geometry
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NW-like interpretation for Schrödinger spacetime ?
G :Schrödinger group is non-semisimple
Killing form is degenerate
Is it possible to apply the NW argument straightforwardly?
Problems
Q1. What is the corresponding coset ?
Q2. What is the symmetric 2-form ?
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Q1. What is the corresponding coset ?
Ans. to Q1.
Physical assumptions
Assump.1
No translation condition.
Assump.2
Lorentz subgroup condition.
doesn’t contain
contains
and
A candidate for the coset
Due to
,
is not contained in the group H
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Q2. What is the symmetric 2-form ?
Ans. to Q2.
NW argument?
It is possible if the coset is reductive
Reductiveness :
EX pp-wave, Bargmann
However, the Schrödinger coset is NOT reductive.
Nappi-Witten argument is not applicable directly.
How should we do?
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The construction of symm. 2-form for the non-reductive case
[Fels-Renner, 2006]
The condition for the symm. 2-form
H-invariance of symm. 2-form
A generalization of NW argument
The indices [m],… are defined up to H-transformation
The group structure const. is generalized.
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Let’s consider the following case:
Structure constants:
M
D
D
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2-form:
vielbeins:
where
coordinate system
metric:
where
(
has been absorbed by rescaling
.)
Similarly, we can derive the metric for the case with an arbitrary z.
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Gravity dual to Lifshitz fixed point
Let’s consider
: a subalgebra of Sch(2)
algebra:
Take
2-form:
metric:
vielbeins:
[Kachru-Liu-Mulligan, 0808.1725]
Unique!
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Lifshitz model (z=2)
2nd order
4th order
scale invariance with z=2
But the symmetry is not Schrödinger.
The theory, while lacking Lorentz invariance, has particle production.
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4.
String theory embedding of
Schrödinger spacetime
[S.A. Hartnoll, K.Y., 0810.0298]
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String theory embedding
Known methods:
1. null Melvin Twist (NMT)
TsT transformation
[Herzog-Rangamani-Ross] [Maldacena-Martelli-Tachikawa] [Adams-Balasubramanian-McGreevy]
2. brane-wave deformation
1. null Melvin Twist
EX
[Hartnoll-K.Y.]
extremal D3-brane
NMT, near horizon
Non-SUSY
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2. brane-wave deformation
[Hartnoll-KY]
Our idea
Allow the coordinate dependence on
the internal manifold X5
Only the (++)-component of Einstein eq. is modified.
The function
has to satisfy
EX
For
we know the eigenvalues:
Thus the spherical harmonics with
The sol. preserves 8 supertranslations (1/4 BPS)
gives a Schrödinger inv. sol.
super Schrödinger symm.
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The solution with an arbitrary dynamical exponent z
Dynamical exponent appears
The differential eq. is
For
case
= a spherical harmonics with
The moduli space of the solution is given by spherical harmonics
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The solution with NS-NS B-field
non-SUSY
The function
By rewriting
Here
has to satisfy the equation
as
is still given by a spherical harmonics with
has been lifted up due to the presence of B-field
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Scalar field fluctuations
S5 part
Laplacian for S5 :
Eigenvalues for S5
Eq. for the radial direction:
The solution:
: large negative
(modified Bessel function)
: pure imaginary (instability)
(while
is real)
The scaling dimension of the dual op. becomes complex
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Characteristics of the two methods
1.
null Melvin Twist
Applicable to the finite temperature case.
Schrödinger BH sols.
Non-SUSY even at zero temperature.
Only for z=2 case (?)
2.
brane-wave deformation
SUSY backgrounds
(at most 8 supertranslations, i.e., 1/4 BPS)
Applicable to arbitrary z
spherical harmonics
Instability
Difficult to apply it to the finite temperature case
Generalization of our work:
[Donos-Gauntlett] [O Colgain-Yavartanoo] [Bobev-Kundu]
[Bobev-Kundu-Pilch] [Ooguri-Park]
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5.
Summary and Discussion
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Summary
1.
Coset construction of Schrödinger spacetime
[Schafer-Nameki-Yamazaki-KY]
Gravity dual for Lifshitz field theory
applicable to other algebras
2.
Supersymmetric embedding into string theory (brane-wave)
[Hartnoll-KY]
Discussion
No concrete example of AdS/NRCFT where both sides are clearly understood.
1.
If we start from gravity
(with the embedding of Sch. algebra)
Difficulty of DLCQ (including interactions)
2.
If we start from the well-known NRCFTs
(with the conventional NR limit)
What is the gravity solution?
NR ABJM
gravity dual
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Thank you!
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DLCQ and deformation
DLCQ (x- -cpt.)
1
pp-wave def.
[Goldberger et.al]
Sch. symm.
2
3
[Son, BM]
2002- in the context of pp-wave
Historical order
-compactification is important for the interpretation as NR CFT
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